A1260005438FlQgW

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A Sparse Grid Collocation Approach to Stochastic
Inverse Problems
NICHOLAS ZABARAS
Materials Process Design and Control
Laboratory Sibley School of Mechanical
Aerospace Engineering 188 Frank H T Rhodes
Hall Cornell University Ithaca, NY 14853 Email
zabaras_at_cornell.edu URL http//mpdc.mae.cornell.e
du/
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CASCADING UNCERTAINTIES IN REAL PHENOMENA
Application
Thermal analysis of functionally graded systems
Only statistical/averaged quantities available to
characterize these structures experimentally.
Eg. Volume fraction, two-point correlation and
semi-variograms
Analysis of oil flow in geological strata
Uncertainties across multiple length and time
scales. Must incorporate them for realistic
analysis
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DESIGN/ESTIMATION IN THE PRESENSE OF UNCERTAINTY
Design thermo-mechanical processes for optimal
performance in the presence of multiple
uncertainties
Estimate location/magnitude of initiation events,
e.g. contamination source detection parameter
identification in the presence of uncertainties,
etc.
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KEY ISSUES WITH ROBUST DESIGN

1. Can ideas from deterministic design and
   optimization be incorporated?
2. Can validated, mature, deterministic models be
   directly used?
3. Can the framework be made scalable? How
   parallelizable can it be?
4. How can experimental information be encoded for
   effective design?

5
SCALABLE STOCHASTIC OPTIMIZATION FRAMEWORK

1. Framework based on finite dimensional
   representation of input uncertainty.
2. Estimation/Design posed as a functional/discrete
   optimization problem
3. Use appropriate stochastic representation forms
   to convert (infinite dimensional) stochastic
   optimization into a finite dimensional
   deterministic optimization
4. Optimization strategy based on repeated solutions
   to the direct problem (comparison with Bayesian)
5. The solution of the stochastic direct problem A
   scalable, embarrassingly parallel strategy that
   allows the use of blackbox deterministic
   simulators

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SOLUTION TO THE STOCHASTIC DIRECT PROBLEM

1. Various sources of uncertainty.
2. Represent each using a finite set of uncorrelated
   random variables
3. The dependant variable lies in the produce space
   of all these variations

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COLLOCATION BASED FRAMEWORK
The stochastic variable at a point in the domain
is an M dimensional function. The GPCE method
approximates this M dimensional function using a
spectral element discretization (which causes the
coupled set of equations)
Spectral discretization represents the function
as a linear combination of sets of orthogonal
polynomials. Instead of a spectral
discretization, use other kinds of polynomial
approximations Use simple interpolation, that is,
sample the function at some points and construct
a polynomial approximation
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COLLOCATION BASED FRAMEWORK
Need to represent this function
Sample the function at a finite set of points
Use polynomials (Lagrange polynomials) to get an
approximate representation
Function value at any point is simply
Stochastic function in 2 dimensions
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COLLOCATION BASED FRAMEWORK
Spatial domain is approximated using a FE, FD, or
FV discretization. Stochastic domain is
approximated using multidimensional interpolating
functions
Once the interpolation function has been
constructed, the function value at any point
is just
Going back to our problem of interest, we want to
construct a polynomial approximation to our
variables (u,p,?)
One can construct the stochastic solution by
solving at the N points specified by
This is just a set of deterministic equations,
with a specified value of uncertainty given by
these M-tuples
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CONSEQUENCES
Construct the complete stochastic solution by
sampling the stochastic space at N distinct
points. That is, solve the deterministic set of
equations defined at the M- tuple

• This results in a completely DECOUPLED set of
  equations. Have to solve N distinct sets of
  equations
• Possibility of parallelization Have to solve
  these equations separately, this makes the
  solution embarrassingly parallel.

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CONSTRUCTING MULTIDIMENSIONAL INTERPOLANTS
Simple tensor products
This quickly becomes impossible to use. For
instance, if M10 dimensions and we use k points
in each direction
Number of points in each direction, k Total number of sampling points
2 1024
3 59049
4 1.05x106
5 9.76x106
10 1x1010
One idea is only to pick the more important
points from the tensor product grid.
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SPARSE GRID COLLOCATION
In higher dimensions, not all the points are
equally important. Some points contribute less to
the accuracy of the solution (e.g. points where
the function is very smooth, regions where the
function is linear, etc.). Discard the points
that contribute less SPARSE GRID COLLOCATION
1D sampling
Tensor product
2D sampling
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SPARSE GRID COLLOCATION
Extensively used in statistical
mechanics Provides a way to construct
interpolation functions based on minimal number
of points Provides a way to progress naturally
from univariate interpolation to multivariate
interpolation. First construct one dimensional
interpolation functions Construct the required
multi-dimensional as a linear combination of
products of these 1D functions How to choose the
degree and coefficients of these
products? Smolyak (1963) came up with a set of
rules to construct such products1.

1. S. Smolyak, Quadrature and interpolation formulas
   for tensor products of certain classes of
   functions, Soviet Math. Dokl., 4 (1963) 240--243.

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SPARSE GRIDS
Number of points 29 Number of points full grid
81
Number of points 145 Number of points full grid
1089
Number of points 69 Number of points full grid
729
Number of points 441 Number of points full grid
35937
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SPARSE GRIDS
The number of sampling points in M dimensions is
given by the cardinality of the set
The plot shows the number of sampling points as a
function of the depth of interpolation for d 2,
3, 5 .
Number of sampling points, N
It is straightforward to come up with bounds on
N1,2.
Depth of interpolation, q

1. M. Babuska, F. Nobile, R. Tempone, A Stochastic
   Collocation Method for Elliptic Partial
   Differential Equations with Random Input Data,
   SIAM J Num. Anal. 45 (2007) 1005-1034
2. V. Barthelmann, E. Novak and K. Ritter, High
   dimensional polynomial interpolation on sparse
   grids, Adv. Comput. Math. 12 (2000) 273288

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ERROR ESTIMATES
Depending on the order of the one-dimensional
interpolant, one can come up with error bounds on
the approximating quality of the interpolant1,2.
If piecewise linear 1D interpolating functions
are used to construct the sparse interpolant, the
error estimate is given by
where N is the number of sampling points,
If 1D polynomial interpolation functions are
used, the error bound is
assuming that f has k continuous derivatives

1. V. Barthelmann, E. Novak and K. Ritter, High
   dimensional polynomial interpolation on sparse
   grids, Adv. Comput. Math. 12 (2000), 273288
2. E. Novak, K. Ritter, R. Schmitt and A.
   Steinbauer, On an interpolatory method for high
   dimensional integration, J. Comp. Appl. Math. 112
   (1999), 215228

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IMPLEMENTATION
Solution Methodology
PREPROCESSING Compute list of collocation points
based on number of stochastic dimensions, d and
level of interpolation, q
Use any validated deterministic solution
procedure. Completely non intrusive
Solve the deterministic problem defined by each
set of collocated points
POSTPROCESSING Compute moments and other
statistics with simple operations of the
deterministic data at the collocated points
Std deviation of temperature Natural convection
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POSTPROCESING

• Once the function values are available at all the
  sparse points, one can construct the stochastic
  solution.
• One is usually interested in the mean, standard
  deviation and the probability density function of
  the dependant variables at various points in the
  physical domain
• Mean and higher moments
• These moments are computed very efficiently
  as a linear combination of the function values at
  the sparse grid points with a set of a priori
  computed weights
• Probability density function (PDF)
• The function is approximated using the
  sparse grid method. Following this a large number
  of function evaluations are performed. A
  normalized histogram of the function distribution
  then gives the PDF

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SOLVING THE STOCHASTIC DIRECT PROBLEM RECAP
Given multiple sources of uncertainty affecting
the system in the form of a function of a finite
set of uncorrelated random variables.
Total dimensionality of the space in which the
dependant variable resides in is
The solution to the stochastic PDE
Is found by solving N deterministic PDEs
The solution is written as
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MATHEMATICAL PRELIMINARIES THE DIRECT PROBLEM
?, a bounded connected region in ?nsd. Boundary
divided into two disjoint surfaces, ??h U ???
Consider IBVP, transient heat diffusion in the
domain Uncertainty in application of thermal BC
as well as uncertainties in coefficients
(material properties)
Coefficients are realizations from a space of
events. Denote this space as ??. Associate a
s-algebra ?? and a probability measure ?? ?? ?
0,1 Results in a probability space (??, ??, ??)
that describes the variability in the coefficients
Can define analogous probability space for
stochastic BC (?q, ?q, ?q)
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MATHEMATICAL PRELIMINARIES PRODUCT SPACE
The temperature in the domain depends on the
uncertainties in the BC as well as uncertainties
in the coefficient Denote this dependence as
Stochastic processes have different structures
with respect to ? and x, require the definition
of tensor product spaces for numerical
approximation1.
Product space Let H1 and H2 be Hilbert spaces. A
basis of the tensor space is
the Cartesian product of the basis of H1 and H2
1,2
The product space is equipped with the inner
product

1. Babuska et. al, Galerkin Finite Element
   Approximations of Stochastic Elliptic Partial
   Differential Equations, SIAM J Num Analysis, 42
   (2004)
2. K. Yoshida, Functional Analysis, 5e (1977)
   Springer-Verlag

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MATHEMATICAL PRELIMINARIES FUNCTION SPACE
Denote the Banach space as the space of
functions that are p-th power integrable
if
Denote the Sobolev space as the space of
functions having generalized derivatives up to
order s in the space . Whenever p2, we use
For a Sobolev space, define its
stochastic counterpart, as the product space
Denote as H(??h ) the space of functions defined
on ??h with finite k-th norm (k any positive
integer)
Define the corresponding stochastic Banach space
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MATHEMATICAL DEFINITION THE DIRECT PROBLEM
Denote the tensor product Hilbert space
Find the stochastic function Such that P-a.e.
(almost everywhere) the following hold
For brevity, we denote the above set of equations
as
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SOME SMOOTHNESS ASSUMPTIONS
ASSUMPTION 1 q is stochastic functions that is
bounded
ASSUMPTION 2 ? is positive and uniformly
coercive
ASSUMPTION 31 The stochastic BC, q satisfies
some regularity condition i.e. for an a priori
defined k
EXISTANCE AND UNIQUENESS With the above
assumptions, the set of equations allows a unique
solution. Direct application of Theorems by
Aronson2 and Eklund3.

1. Ch. Schwab, R. A. Todur, Sparse finite elements
   for Stochastic elliptic problems - Higher
   moments, Computing 71 (2003) 43-63
2. D.G. Aronson, Non-negative solutions of linear
   parabolic equations, Ann. Scuola Norm. Sup. Pisa
   22 (1968) 607-694
3. N.A.Eklund, Existance and representation of
   solutions of parabolic equations, Proc. Amer.
   Math. Society 47 (1975) 137142

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THE STEADY STATE DIRECT PROBLEM
Denote the tensor product Hilbert space
Find the stochastic function Such that P-a.e.
(almost everywhere) the following hold
EXISTENCE AND UNIQUENESS With the assumptions,
this set of equations allows a unique solution.
Direct application of Lax-Milgram lemma.

1. Babuska et. al, Galerkin Finite Element
   Approximations of Stochastic Elliptic Partial
   Differential Equations, SIAM J Num Analysis, 42
   (2004)

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THE STOCHASTIC INVERSE PROBLEM INTUITIVE IDEAS
The solution to the direct problem assumes that a
neat representation of the input uncertainties in
terms of a finite number of uncorrelated random
variables is available
?(a)
In the inverse problem, we are given the
stochastic dependant variable and are asked to
compute the input stochastic processes that
result in the given dependant variable
?-1(a)
But what is actually given?
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THE STOCHASTIC INVERSE PROBLEM INTUITIVE IDEAS
Issues

• The measured/desired quantity is a stochastic
  process that depends on multiple sources of
  uncertainty. How to isolate the effect of one
  process?
• Physically motivated issues In most
  applications, one measures realizations and
  subsequently provides ONLY moments and/or a PDF
  of the dependant variable

?(a)
Source A
HOW?
Source B
?-1(a)
Source C
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THE STOCHASTIC INVERSE PROBLEM INTUITIVE IDEAS
Given limited information about the dependant
stochastic variable- in terms of ONLY, say, k
moments or a PDF is it possible to construct a
stochastic representation of the design variables?
For clarity, consider the direct problem again.
Moments, PDF
?(a)
Suppose that we are ONLY given moments or a PDF
of the input stochastic process, what can be said
about the dependant stochastic process
Results in ambiguity in the exact nature of the
input stochastic process
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POSING THE STOCHASTIC INVERSE PROBLEM
Given this ambiguity, can anything be said about
the dependant stochastic process?
Limit ourselves to the case of a stochastic
elliptic equation Pose the direct problem in the
variational (weak) form. Smoothness conditions of
the coefficients and load1,2,3
Theorem 1 The stochastic elliptic problem has a
unique solution (in the expectation sense)
EXISTENCE AND UNIQUENESS With the assumptions,
this set of equations allows a unique solution.
Direct application of Lax-Milgram lemma.

1. Ch. Schwab, R. A. Todur, Sparse finite elements
   for Stochastic elliptic problems - Higher
   moments, Computing 71 (2003) 43-63
2. I. Babuska, R. Tempone, G. E. Zouraris, Galerkin
   finite elements approximation of stochastic
   finite elements, SIAM J. Numer. Anal. 42 (2004)
   800825.
3. I. M. Babuska, F. Nobile, R. Tempone, A
   stochastic collocation method for elliptic
   partial differential equations with random input
   data, SIAM Journal on Numerical Analysis 45
   (2007) 1005-1034

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POSING THE STOCHASTIC INVERSE PROBLEM
But what about the expectation and higher order
moments
Theorem 2 Under certain smoothness conditions
for the input stochastic process, the k-th order
moment of solution exists and is unique The k-th
order moment or the k-order correlation exists in
?k. Construct anisotropic stochastic Sobolov
spaces1 on ?x?xx? as a tensor product
Construct the appropriate operators in the
anisotropic tensor product space
Products of bounded positive homeomorphisms
between Hilbert spaces induce corresponding
homeomorphisms between tensor products of these
spaces. Application of the Lax-Milgram lemma for
uniqueness

1. Ch. Schwab, R. A. Todur, Sparse finite elements
   for Stochastic elliptic problems - Higher
   moments, Computing 71 (2003) 43-63
2. I. Babuska, R. Tempone, G. E. Zouraris, Galerkin
   finite elements approximation of stochastic
   finite elements, SIAM J. Numer. Anal. 42 (2004)
   800825.
3. I. M. Babuska, F. Nobile, R. Tempone, A
   stochastic collocation method for elliptic
   partial differential equations with random input
   data, SIAM Journal on Numerical Analysis 45
   (2007) 1005-1034

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POSING THE STOCHASTIC INVERSE PROBLEM
Physically meaningful designed quantity would be
moments and/or a PDF of the corresponding input
stochastic process
?-1(a)
?-1(a)
Given measurements in terms of PDF/moments gt
Construct the optimal PDF/moments of input
quantities
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PROBLEM STATEMENT
For the sake of generality, we consider the
stochastic parabolic problem (transient heat
conduction), though some of our proofs are for
the stochastic elliptic problem (heat conduction).
??h
?
???
sensors
Input uncertainties Thermal conductivity
(known), heat flux (unknown) Given PDF or
moments of temperature at some sensor locations
TYPE I Find the moments of the stochastic heat
flux, such that the solution of the system
defined by the above SPDE results in a
temperature profile whose k moments match the
given experimental data at s sensor locations
TYPE II Find the PDF of the stochastic heat
flux, such that the solution of the system
defined by the above SPDE results in a
temperature profile whose PDF matches the given
experimental data at s sensor locations
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THE DESIGN PROBLEM
The unknown stochastic heat flux can be written
in terms of a finite set of uncorrelated random
variables. Following the usual procedure in
collocation based solution techniques, the
unknown stochastic heat flux is written in terms
of nq deterministic realizations of the heat flux
Have to design these nq deterministic functions.
Pose problems in terms of designing these
functions
TYPE I Designing Moments Find q(x,t,?i),
i1,,nq such that the solution of the system
defined by the above SPDE results in a
temperature profile whose k moments match the
given experimental data at s sensor locations
TYPE II Designing PDF Find q(x,t,?i), i1,,nq,
such that the solution of the system defined by
the above SPDE results in a temperature profile
whose PDF matches the given experimental data at
s sensor locations
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THE OPTIMIZATION PROBLEM
Assuming that a solution exists in the sense of
Tikhonov, look for a solution
is a cost functional that quantifies
how well the designed heat flux performs. Convert
the design problem into an optimization problem.
Problem Type I Define cost functional in terms
of difference in moments
Computed moments
Measured/desired moments
Problem Type II Define cost functional in terms
of difference in PDF
is the inverse CDF (chosen for
computational efficiency and known support1 0
1 )

1. N. Zabaras and B. Ganapathysubramanian, A
   scalable framework for the solution of stochastic
   inverse problems using a sparse grid collocation
   approach, J. Computational Physics, submitted
   (Preprint)

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THE OPTIMIZATION PROBLEM
Represent the cost functional in terms of the nq
deterministic functions. Utilize the sparse grid
representation to simplify the cost
functional. Using the sparse representation of
the dependant variable
Represent the moments as
And the inverse cumulative distribution function
as
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GRADIENT OF THE COST FUNCTIONAL
Gradient based optimization utilized. Need to
define the gradient of the cost functional.
Define the notion of the directional derivative
After some manipulations, can write the gradient
as
Problem Type I
Problem Type II
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STOCHASTIC SENSITIVITY EQUATIONS
Directional derivative term inside the gradient
can be simplified to
Where T is the linear in ?qv part of the
temperature
Thus, the calculation of the gradient of the cost
functional requires the computation of the
stochastic sensitivity of the temperature. But
stochastic sensitivity definition very simple
using sparse grid collocation
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STOCHASTIC SENSITIVITY EQUATIONS
The stochastic sensitivity equations are simply
directional derivatives of the N deterministic
equation w.r.t. each of the design parameters qi.
The stochastic sensitivity equations are
The stochastic sensitivity can then be written in
terms of solution to these Nn equations as
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THE OPTIMIZATION FRAMEWORK

• The stochastic optimization framework uses
• A minimal set of decoupled, deterministic direct
  problems
• A minimal set of decoupled, deterministic
  sensitivity problems
• The only serial part of this framework is the
  computation of the cost functional and the
  gradient gt highly scalable

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NUMERICAL TESTS
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1D IHCP Test 1 Problem Type I- Reconstructing
moments
1D rod of length L1.0
The thermal conductivity and the heat flux are
stochastic. The thermal conductivity, k varies as
The moments of the temperature at the sensor
location are given. These experimental moments
are computed by running a direct problem using a
predefined distribution for the stochastic heat
flux. A standard Beta distribution with p1.5 and
q 2 is used Run direct problem using higher
discretization in physical space and stochastic
space than corresponding inverse problem- prevent
inverse crimes
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1D IHCP Test 1 Reconstructing moments
Convergence with increasing depth of interpolation
depth Design variables
2 5
4 17
6 65
8 257
Accuracy increases with depth
Use one r.v. to represent q. ? U0,1 First few
(7) moments captured with very coarse
representation of the space
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Hierarchical stochastic optimization A
rudimentary example
The optimization with lower depth of
interpolation is faster Utilize this to construct
a coarse stochastic solution. Use this solution
as an initial guess to run a refined optimization
problem
Similar in idea to accelerated convergence in
multi-grid methods We call this hierarchical
stochastic optimization method. Results in
significant reduction in computational effort
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1D IHCP Test 1b Reconstructing moments
Invariance with choice of the representation of q
Use one random variable used to represent q. 4
runs with different r.v. and associated
probability measures were performed
N Exp Beta Gamma Normal Uniform
1 0.42858 0.42855 0.42864 0.42854 0.42856
2 0.23811 0.23802 0.23779 0.23835 0.23882
3 0.15155 0.15193 0.15180 0.15177 0.14919
4 0.10494 0.10533 0.10538 0.10503 0.10572
5 0.07698 0.07698 0.07724 0.0768 0.07764
6 0.05889 0.05831 0.05879 0.05834 0.06095
Almost identical results using all four different
representations
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1D IHCP Test 2 Problem Type II- Reconstructing
PDFs
1D rod of length L1.0
The thermal conductivity and the heat flux are
stochastic. The thermal conductivity, k varies as
The PDF of the temperature at the sensor location
is given. These experimental moments are
computed by running a direct problem using a
predefined distribution for the stochastic heat
flux. CASE 1 Finite support A standard Beta
distribution with p1.5 and q 2 is used CASE 2
Unbounded support A standard Normal distribution
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1D IHCP Test 2 Reconstructing PDFs
Convergence with increasing depth of
interpolation Finite support
Use one random variable to represent q. ? U0,1
For lower depths of interpolation, the tails are
not captured well. Slightly pixilated
representation for medium depths of
interpolation Almost perfect representation for
higher depths of interpolation
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1D IHCP Test 2 Reconstructing PDFs
Convergence with increasing depth of
interpolation Infinite support
Use one random variable to represent q. ? U0,1
For lower depths of interpolation, the tails are
not captured well. Slightly pixilated
representation for medium depths of
interpolation Almost perfect representation for
higher depths of interpolation
48
1D IHCP Test 2b Reconstructing PDFs
Invariance with choice of the representation of q
Use one r.v. to represent q. 4 runs with
different r.v. and associated probability
measures were performed. Used Uniform random
variable U0,1 Normal random variable
N(0,1) Gamma random variable G(2) Beta random
variable B(1.5,2)
49
1D IHCP Test 3 Reconstructing PDFs
Reconstructing high dimensional input stochastic
processes
The thermal conductivity and the heat flux are
stochastic. The thermal conductivity, k varies as
The PDF of the temperature at the sensor location
is given. These experimental moments are
computed by running a direct problem using a
multidimensional stochastic representation of q
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1D IHCP Test 3 Reconstructing PDFs
Reconstructing high dimensional input stochastic
processes
Convergence w.r.t. depth of interpolation. Use
one random variable to represent q.
Comparison of reconstructed and original
stochastic heat flux. Circles represent
experimental PDF, line is the reconstructed PDF
51
2D IHCP Transient Analysis
Design q
See if framework is applicable to solve transient
problems Consider a 2D domain -0.5 0.52 The
temperature on the right wall is held
constant The thermal conductivity is stochastic.
T 0.5
Variable conductivity
The thermal conductivity has a mean value of 20
and is correlated with a correlation kernel that
is exponential. The correlation length is
0.1. Represent the conductivity using a finite
set of uncorrelated random variables- Karhunen
Loeve expansion
Use first three random variables to represent
(known) variability in k
52
2D IHCP Problem definition and experimental
data
Design q
Given the PDF of temperature at sensor locations
over the time interval 0, T , reconstruct the
PDF of heat flux applied on the left boundary in
0,T
Variable conductivity
T 0.5
Sensors are located at a distance 0.1L into the
domain
Constructing the PDF of temperatures at the
sensors A known decaying stochastic heat flux
profile is applied on the left boundary. The
heat flux is assumed to have an exponential
correlation with b 0.5 Represent the
experimental heat flux using a finite set of
uncorrelated random variables. Use the first
three eigen values
53
2D IHCP Problem definition and experimental
data
Output random process, Temp
Input random process, flux
Making sure inverse crimes are not committed.
Very high discretization spatially, temporally
and in stochastic space.
54
2D IHCP Optimization
Spatial discretization 40x40 quad elements Time
discretization 50 time steps, dt
0.001 Representation of k Level 5 sparse grid
441 points in 3 dim stochastic space
Design each nodal dof on the boundary for each
time 4150 stochastic quantities
Each q(y,t,?) represented using one uniform
random variable. Use sparse grid interpolants to
represent these
Set N 65. Corresponds to level 6
interpolation Total number of design variables
N4150 133,250 Each direct stochastic problem
requires solution of 65441 28665 deterministic
problems Each direct problem corresponds to
414150 84,050 variables Total dof/iteration
2.41 billion
55
2D IHCP Results
Experimental flux PDF
Reconstructed flux PDF
Use 40 nodes of local linux cluster 160
processors Each iteration took about 74
minutes. Movies compare the time evolution of
reconstructed PDF and the experimental PDF at two
locations
Endpoint, y -0.5
Midpoint, y 0.0
56
2D IHCP Scalability/Speedup
Run the problem using different number of
processes Estimate the parallelizability of the
framework by looking at time taken to complete
one iteration in the optimization problem
Number of processor almost inversely proportional
to the time taken. Sequential part is negligibly
small. Very highly scalable framework
57
Robust design A 3D example
Most of the critical components in many
applications/devices are usually fabricated from
polycrystalline/functionally graded/heterogeneous
materials
The properties at the device-scale (thermal and
electrical conductivity, elastic moduli and
failure mechanisms) depend on the microstructure
and material distribution at the meso-scale
Heat sinks in microelectronic devices
But it is not possible to completely characterize
the microstructure distribution of a device.
Have to design the operating conditions/parameters
taking into account this limited information
about the microstructure.
Robust design in the presence of topological
uncertainty
Thermal tiles in aerospace structures
58
Robust design Problem definition
Given limited topological information about the
microstructure, design the optimal (stochastic)
heat flux to be applied on one end of the device
such that a required (stochastic) temperature is
maintained at the other end.
Limited information in the form of a single
experimental image of the microstructure. A
Silver-Tungsten metal-metal composite
2a

• Desired temperature profile
• Mean temperature is a linearly varying profile
• Standard deviation of 0.05
• Choose a normal distribution N(Tmean. Tstd)

Maintain temperature here
Design flux here
Topology/material distribution unknown
Statistical descriptors available
59
Robust design Problem definition

• Design criterion
• Maintain a specified thermal profile on the
  right wall
• This thermal profile is given in terms of a PDF

• Additional uncertainties
• Do not know the exact microstructure that the
  device is made up of.
• Only know certain statistical correlations that
  the microstructure satisfies.
• Consider the microstructure to be a random field
  ?.

• Design variables
• Must design the PDF of the optimal heat flux

60
Robust design Problem definition
Dimension reduction methodology applied to the
problem of representing the microstructure
variability. Results in a 9 dimensional
stochastic space. This space represents the set
of all microstructures satisfying the
experimentally determined statistics (volume
fraction and two point correlation)
Robust design to account for this uncertainty
Physical domain 20 µm x 20µm x
20µm Computational domain 65x65x65 Microstructure
stochastic space 1177 collocation points The
design stochastic heat flux represented as a
Bezier surface number of parameters 25 The
design heat flux is stochastic- belongs to a
stochastic space The temperature depends on the
uncertainty in the heat flux and the
microstructural uncertainty
One realization of the microstructure
Experimental statistics
61
Robust design Results
Run optimization problem on 46 nodes of our Linux
cluster Each iteration 9 hours Solve 117765
76505 deterministic problems in each
iteration Each deterministic problem has 65x65x65
275625 DOF Total number of design variables
6555 1625 Total dof/iteration 21.25 billion
62
Robust design First/second order statistics
Bounds on the flux
Mean heat flux surface
The mean value of the optimal heat flux The
figure on the right shows the bounds on the heat
flux (Mean /- one standard deviation)
63
Robust design PDFs of heat flux and temperature
Left, PDF of designed optimal heat flux at three
points Right, PDF of resulting temperatures at
three points
64
Temperature statistics on the boundary
Check to see if the designed optimal stochastic
heat flux produces a temperature that satisfies
the design requirements
Left The standard deviation of the temperature.
The standard deviation is less then 0.05 in the
complete region Right The mean temperature on
the right boundary. There is a decrease in the
temperature along the z axis
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Convergence
Convergence with increasing depth of interpolation
Level 7
Level 6
Level 8
Bottom Bounds on the flux
Top Mean heat flux surface
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Convergence
Convergence with refining the Bezier
representation
Bounds on the flux
Mean heat flux surface
Solve optimization problem with two different
Bezier representations
5x5
(5,5) representation 3175 Design variables
(9,9) representation 10287 Design variables
9x9
Less than 0.1 variation in solutions
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Conclusions
Developed a scalable framework for
estimation/design problems in the presence of
uncertainty Framework provides an attractive
alternative to Bayesian based methods. Does not
require an assumption of the prior Framework is
based on ideas from stochastic sparse grid
representation of stochastic processes and
deterministic optimization principles Seamlessly
allows the linking of any validated deterministic
blackbox simulator. Significant ramifications in
processes that are affected by multiple sources
of uncertainty. Future prospects Application to
deformation process design, multiscale estimation
problems